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Functions defined by parts
Indicate the domain and the image of the following function:
$$f(x)=\left\{\begin{array}{rcl} -1 & \mbox{ if } & x<-1 \\ 2x+1 & \mbox{ if } & -1\leq x < 2 \\ 2 & \mbox{ if } & x\geq 3\end{array}\right.$$
We can find the domain of the first function from the intervals in its definition:
$$Dom (f) = (-\infty,-1)\cup[-1,2)\cup[3,+\infty)=(-\infty,2)\cup[3,+\infty)$$
To determine the image we can concentrate on the images of the different functions that compose the function, bearing in mind the domain where they are defined.
For $x < -1$ or $x > 3$ we have no problems since we know the valuation of the function in these intervals.
For the straight line between $-1$ and $2$, we calculate the valuation in the above mentioned points:
$2x+1$ in $x =-1$ values $-1$
$2x +1$ in $x = 2$ values $5$
Therefore $Im (f) = [-1, 5)$.
It is necessary to bear in mind that we will include the extreme points in the image depending on whether they are included or not in the definition of the function.
$Dom(f)=(-\infty,2)\cup[3,+\infty)$, $Im (f) = [-1, 5)$